My first step in doing limits is to sub in the approach value, this will give me Lim arctan (0/0)
This just about guarantees that the algebraic expression will factor and the offending factor will cancel.
Sure enough
lim arctan[(x^2 - 4)/(3x^2-6x)] x->2
= lim arctan[(x+2)(x-2)]/[3x(x-2)]
=lim arctan[(x+2)]/[3x]
= lim arctan(2/3)